Quadratic Equations

To solve the quadratic equation $ax^{2}+bx+c=0$ , we factorise the quadratic expression $ax^{2}+bx+c$ into two factors. The solutions are then obtained by setting each of the factors equal to zero. We shall obtain two solutions.

Let’s look at some examples.

Example

Solve the equations:

i) $(2x-1)(3x+2)=0$

ii) $2x^{2}+x-3=0$

i) $(2x-1)(3x+2)=0$

Either $2x-1=0$

$x = \frac{1}{2}$

Or, $3x+2=0$

$x = \frac{-2}{3}$

ii) $2x^{2}+x-3=0$

$2x^{2}+3x-2x-3=0$

$x(2x+3)-1(2x+3)=0$

$(x-1)(2x+3)=0$

So, either $x-1=0$

$x=1$

Or, $2x+3=0$

$x = \frac{-3}{2}$

Example

Solve the quadratic equation:

$x^{2}-16=0$

$x^{2}=16$

$x = \sqrt{16}$

$x=\pm 4$

Or, we can do this by factors

$x^{2}-16=0$

$(x+4)(x-4)=0$

Either $x+4=0$

$x=-4$

Or, $x-4=0$

$x=4$

Example

Solve the equation $6x+7 = \frac{5}{x}$

$6x+7 = \frac{5}{x}$

We multiply both sides by x.

$x(6x+7)=5$

Then, we expand the bracket and simplify.

$6x^{2}+7x=5$

$6x^{2}+7x-5=0$

Factorise: $6x^{2}+10x-3x-5=0$

$2x(3x+5)-1(3x+5)=0$

$(2x-1)(3x+5)=0$

Solve: $x = \frac{1}{2}$ or $x = \frac{-5}{3}$

The quadratic expression $ax^{2}+bx+c$ is not always factorisable in rational numbers. This situation arises when the quantity $b^{2}-4ac$ , called the discriminant, is not a perfect square, meaning it has no rational square roots.

We then solve the quadratic equation $ax^{2}+bx+c=0$ , using the formula $x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

Consider the quadratic equation:

$2x^{2}-7x+1=0$

For the equation:

$a = 2$ , $b = -7$ and $c = 1$ .

We substitute these numbers into the formula above. So:

$x=\dfrac{-\left( -7\right) \pm \sqrt{\left( -7\right) ^{2}-4\left( 2\right) \left( 1\right) }}{2\left( 2\right) }$

$=\dfrac{7\pm \sqrt{49-8}}{4}$

We obtain $x = \frac{7- \sqrt{41}}{4}$

$= 0.49$ (3 significant figures)

Or, $x = \frac{7 + \sqrt{41}}{4}$

$= 3.35$ (3 significant figures)

Let’s look at some examples.

Example

Solve the quadratic equation:

$5x^{2}-14x+3=0$

Giving answers correct to three significant figures

$5x^{2}-14x+3=0$

$a=5, b=-14, c=3$

$x=\dfrac{14\pm \sqrt{14^{2}-4\left( 5\right) \left( 3\right) }}{2\left( 5\right) }$

$=\dfrac{14\pm \sqrt{136}}{10}$

$x=0.234$ or $x=2.57$ , each correct to three significant figures.

Example

Find the possible values of x for which:

$2x^{2}-7x+3=0$

Here, $a=2$ , $b=-7$ and $c=3$ .

We substitute these numbers into $x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

$x=\dfrac{-\left( -7\right) \pm \sqrt{\left( -7\right) ^{2}-4\left( 2\right) \left( 3\right) }}{2\left( 2\right) }$

$=\dfrac{7\pm \sqrt{49-24}}{4}$

$=\dfrac{7\pm \sqrt{25}}{4}$

$=\dfrac{7\pm 5}{4}$

$x = \frac{7-5}{4} = \frac{1}{2}$

Or, $x = \frac{7+5}{4} = 3$

The possible values of x are $\frac{1}{2}$ and 3.

Note: In fact, the quadratic equation $2x^{2}-7x+3=0$ can be solved using factorisation.

$2x^{2}-7x+3=0$

$2x^{2}-6x-x+3=0$

$2x(x-3)-1(x-3)=0$

$(2x-1)(x-3)=0$

Either $2x-1=0$ , giving $x = \frac{1}{2}$

Or, $x-3=0$ , giving $x=3$ .

Example

A solid cuboid of length x metres has a square cross-section of side $(x-3)$ metres. Write down, in terms of x, expressions for:

i) The surface are of the square cross-section.

ii) The surface area of one of the longer faces of the cuboid.

Given that the total surface area of the cuboid is 39 square metres, write down an equation in x and show that it reduces to:

$2x^{2}-8x-7=0$

Find the two values of x which satisfies this equation, giving your answers correct to three significant figures. Hence right down the dimensions of the cuboid.

i) Surface are of the cross-section $= (x-3)(x-3)$

$= (x^{2}-6x+9)\:m^{2}$

ii) Surface area of one of the longer faces

$= x(x-3)$

$= (x^{2}-3x)\:m^{2}$

Total surface area of the cuboid:

$= 2(x^{2}-6x+9)+4(x^{2}-3x)$

$= 2x^{2}-12x+18+4x^{2}-12x$

$= 6x^{2}-24x+18$

Given the total surface area $= 39\:m^{2}$ :

$6x^{2}-24x+18=39$

$6x^{2}-24x+18-39=0$

$6x^{2}-24x-21=0$

Dividing by the common factor 3, we obtain:

$2x^{2}-8x-7=0$ , as required.

We use the formula to find x

$x=\dfrac{8\pm \sqrt{64-4\left( 2\right) \left( -7\right) }}{2\left( 2\right) }$

$x=\dfrac{8\pm \sqrt{120}}{4}$

$x=-0.739$ or $x=4.74$ , each to three significant figures.

The dimensions of the cuboid are 4.74 m, 1.74 m

Example

Solve the equation:

$\frac{x-5}{x-2}+1=\frac{3x}{x-1}$

We can simplify the left hand side, then cross-multiply.

$\frac{x-5}{x-2}+1=\frac{x-5+1(x-2)}{x-2}$

$=\frac{x-5+x-2}{x-2}$

$=\frac{2x-7}{x-2}$

So, $\frac{2x-7}{x-2}=\frac{3x}{x-1}$

We now cross-multiply and expand:

$(2x-7)(x-1)=3x(x-2)$

$2x^{2}-9x+7=3x^{2}-6x$

Transposing: $3x^{2}-2x^{2}-6x+9x-7=0$

$x=\dfrac{-3\pm \sqrt{9-4\left( 1\right) \left( -7\right) }}{2\left( 1\right) }$

$x=\dfrac{-3\pm \sqrt{9+28}}{2}$

$x=\dfrac{-3\pm \sqrt{37}}{2}$

Either $x=\dfrac{-3 - \sqrt{37}}{2}$

$= -4.54$

$x=\dfrac{-3 + \sqrt{37}}{2}$

= 1.54

to three significant figures.

Example – Full exam question

(a) Find the value of:

$\dfrac{a+\sqrt{a^{2}+b^{2}}}{a^{2}-2ab}$

When $a=-4$ and $b=-3$ , give your answer as a fraction in its lowest term.

(b) Expand the brackets and simplify:

$(3x^{2}-1)(2x+3)-x(9x-2)$

(c)(i) Factorise $9x^{2}+5x-4$

(ii) Use your answer to part (c)(i) to solve the equation $9x^{2}+5x-4=0$

(d) The sum of three consecutive even numbers is 78. Find these three numbers.

(a) $\dfrac{a+\sqrt{a^{2}+b^{2}}}{a^{2}-2ab}$

When $a=-4$ , $b=-3$ , we obtain:

$\dfrac{-4+\sqrt{\left( -4\right) ^{2}+\left( -3\right) ^{2}}}{}\\ \left( -4\right) ^{2}-2\left( -4\right) \left( -3\right)$

$= \dfrac{-4+\sqrt{16+9}}{16-24}$

$= \dfrac{-4+\sqrt{25}}{-8}$

$= \dfrac{-4+5}{-8}$

$= -\dfrac{1}{8}$

(b) $(3x^{2}-1)(2x+3)-x(9x-2)$

$= 6x^{3}+9x^{2}-2x-3-9x^{2}+2x$

$= 6x^{2}-3$

(c)(i) $9x^{2}+5x-4$

Find two numbers, sum = 5, product= −36.

1┃36

2┃18

3┃12

4┃9 → Good, choose −4,9.

$9x^{2}+5x-4=9x^{2}-4x+9x-4$

$=x(9x-4)+1(9x-4)$

$= (x+1)(9x-4)$

(ii) $9x^{2}+5x-4=0$

$(x+1)(9x-4)=0$

Either $x+1=0$

$= x = -1$

$9x-4=0$

$x=\frac{4}{9}$

(d) Let the three even numbers be:

$2x$ , $2x+2$ and $2x+4$ , as they are consecutive.

Now, $2x+2x+2+2x+4=78$

$6x+6=78$

$6x=72$

$x=12$

So that the three even numbers are:

$2(12), 2(12)+2, 2(12)+4$

$= 24, 26, 28$

Example – Exam question

The distance between London and York is 320 km. A train takes x hours to travel between London and York.

(a) Write down an expression, in terms of x, for the average speed of the train in km/h.

(b) A car takes 2 \frac{1}{2} hours longer than a train to travel between London and York.

The average speed of the train is 80 km/h greater than the average speed of the car.

Form an equation in x and show that it simplifies to $2x^{2}+5x-20=0$ .

(d) Hence find the average speed of the car correct to the nearest km/h.

(a) London → York = 320 km

Average speed $= \frac{Total distance}{Total time}$

Average speed of the train $= \frac{320 km}{x hr}$

$= \frac{320}{x} km/h$

(b) Average speed of the car $= \frac{320}{x+\frac{5}{2}} km/h$

Average speed of the train − average speed of the car = 80

$\frac{320}{x}-\frac{320}{x+\frac{5}{2}} = 80$

Divide by 80 throughout the equation:

$\frac{4}{x}-\frac{4}{x+\frac{5}{2}}=1$

Multiply by $x(x+\frac{5}{2}$ on both sides

$4(x+\frac{5}{2}) - 4x = 1 \times x(x+\frac{5}{2})$

$4x+10-4x = x(x+\frac{5}{2})$

$10=x^{2}+\frac{5}{2}x$

Multiply by 2 on both sides

$20 = 2x^{2}+5x$

Rearranging gives $2x^{2}+5x-20=0$

$x=\dfrac{-5\pm \sqrt{25-4\left( 2\right) \left( -20\right) }}{2\left( 2\right) }$

$=\dfrac{-5\pm \sqrt{185}}{4}$

Either $x=\dfrac{-5-\sqrt{185}}{4}$

$= -4.65$ (two decimal places)

$x=\dfrac{-5+\sqrt{185}}{4}$

$= 2.15$ (two decimal places)

(d) Average speed of the car $= \frac{320}{x+\frac{5}{2}}$

$= \frac{320}{2.15037+2.5}$

$= \frac{320}{4.65037}$

$=68.8$ km/h

69 km/h (nearest km/h)

Example

Solve the equation:

$(2x+1)(3x-2)=49$

We first need to rewrite the equation into the standard form $ax^{2}+bx+c=0$

$(2x+1)(3x-2)=49$

$6x^{2}-4x+3x-2-49=0$

$6x^{2}-x-51=0$

$6x^{2}-18x+17x-51=0$

$6x(x-3)+17(x-3)=0$

$(6x+17)(x-3)=0$

Either $6x+17=0$

$x=\frac{-17}{6}$

$x-3=0$

$x=3$

Previous Topic

Back to Lesson

Next Topic

Last updated: April 15, 2022

This Privacy Policy describes Our policies and procedures on the collection, use and disclosure of Your information when You use our services. It also tells you about your privacy rights and how the law protects you.

We use Your Personal data to provide and improve our services. By using our services, you are agreeing to the collection and use of information in accordance with this Privacy Policy.

Definitions

For the purposes of this Privacy Policy:

• Account means a unique account created for You to access our Service or parts of our Service.

• Company (referred to as either “the Company”, “We”, “Us” or “Our” in this Agreement) refers to Shalom Education Ltd, 86 London Road, (Kingsland Church), Colchester, Essex, CO3 9DW. For the purpose of the GDPR, the Company is the Data Controller.

• Cookies are small files that are placed on Your computer, mobile device or any other device by a website, containing the details of Your browsing history on that website among its many uses.

• Country refers to the United Kingdom.

• Data Controller, for the purposes of the GDPR (General Data Protection Regulation), refers to the Company as the legal person which alone or jointly with others determines the purposes and means of the processing of Personal Data.

• Device means any device that can access the Service such as a computer, a mobile phone or a digital tablet.

• Personal Data is any information that relates to an identified or identifiable individual. For the purposes of GDPR, Personal Data means any information relating to you such as a name, an identification number, location data, online identifier or to one or more factors specific to the physical, physiological, genetic, mental, economic, cultural or social identity.

• Service refers to the use of our website, resources or our tutoring services.

• Service Provider means any natural or legal person who processes the data on behalf of the company. It refers to third-party companies or individuals employed by the company to facilitate the services. This is to provide the services on behalf of the company, to perform services related to the main services or to assist the company in analysing how the service are used. For the purpose of the GDPR, Service Providers are considered Data Processors.

• Usage Data refers to data collected automatically, either generated by the use of the the services or from the service infrastructure itself (for example, the duration of a page visit).

• Website refers to Shalom Education, accessible from https://www.shalom-education.com

• You means the individual accessing or using the Service, or the company, or other legal entity on behalf of which such individual is accessing or using the Service, as applicable.Under GDPR (General Data Protection Regulation), You can be referred to as the Data Subject or as the User as you are the individual using the Service.

Collecting and Using Your Personal Data

Types of Data Collected

Personal Data
Whilst using Our Service, We may ask you to provide us with certain personally identifiable information that can be used to contact or identify you. Personally identifiable information may include, but is not limited to:

• Email address
• First name and last name
• Phone number
• Address, State, Province, ZIP/Postal code, City
• Usage Data

Usage Data

Usage Data is collected automatically when using the Service.

Usage Data may include information such as your Device’s Internet Protocol address (IP address), browser type, browser version, the pages of our services that you visit, the time and date of your visit, the time spent on those pages, unique device identifiers and other diagnostic data.

When You access the Service by or through a mobile device, We may collect certain information automatically, including, but not limited to, the type of mobile device you use, your mobile device unique ID, the IP address of your mobile device, your mobile operating system, the type of mobile Internet browser you use, unique device identifiers and other diagnostic data.

We may also collect information that your browser sends whenever you visit our website or when you access our website by or through a mobile device.

Tracking Technologies and Cookies
We use Cookies and similar tracking technologies to track the activity on our website and store certain information. Tracking technologies used are beacons, tags, and scripts to collect and track information and to improve and analyse our services. The technologies We use may include:

• Cookies or Browser Cookies. A cookie is a small file placed on your Device. You can instruct your browser to refuse all Cookies or to indicate when a Cookie is being sent. However, if you do not accept Cookies, you may not be able to use some parts of our Service. Unless you have adjusted your browser setting so that it will refuse cookies, our services may use cookies.

• Flash Cookies. Certain features of our services may use local stored objects (or Flash Cookies) to collect and store information about your preferences or your activity on our services. Flash Cookies are not managed by the same browser settings as those used for browser cookies. For more information on how you can delete flash cookies, please read “Where can I change the settings for disabling, or deleting local shared objects?” available at https://helpx.adobe.com/flash-player/kb/disable-local-shared-objects-flash.html#main_Where_can_I_change_the_settings_for_disabling__or_deleting_local_shared_objects_

• Web Beacons. Certain sections of our Service and our emails may contain small electronic files known as web beacons (also referred to as clear gifs, pixel tags, and single-pixel gifs) that permit the company, for example, to count users who have visited those pages or opened an email and for other related website statistics (for example, recording the popularity of a certain section and verifying system and server integrity).
Cookies can be “persistent” or “session” Cookies. Persistent Cookies remain on Your personal computer or mobile device when you go offline, whilst Session cookies are deleted as soon as you close your web browser.

We use both session and persistent cookies for the purposes set out below:

• Necessary / Essential Cookies
Type: Session cookies

Administered by: Us

Purpose: These Cookies are essential to provide you with services available through the website and to enable You to use some of its features. They help to authenticate users and prevent fraudulent use of user accounts. Without these Cookies, the services that you have asked for cannot be provided, and we only use these cookies to provide you with those services.

• Cookies Policy / Notice Acceptance Cookies
Type: Persistent cookies

Administered by: Us

Purpose: These Cookies identify if users have accepted the use of cookies on the Website.

• Functionality Cookies
Type: Persistent cookies

Administered by: Us

Purpose: These cookies allow us to remember the choices you make when you use the website, such as remembering your login details or language preference. The purpose of these cookies is to provide you with a more personal experience and to avoid you having to re-enter your preferences every time you use the website.

• Tracking and Performance Cookies
Type: Persistent cookies

Administered by: Third-Parties

Purpose: These cookies are used to track information about traffic to the website and how users use the website. The information gathered via these cookies may directly or indirectly identify you as an individual visitor. This is because the information collected is typically linked to a pseudonymous identifier associated with the device you use to access the website. We may also use these cookies to test new pages, features or new functionality of the website to see how our users react to them.

• Targeting and Advertising Cookies
Type: Persistent cookies

Administered by: Third-Parties

Purpose: These cookies track your browsing habits to enable us to show advertising which is more likely to be of interest to you. These Cookies use information about your browsing history to group you with other users who have similar interests. Based on that information, and with our permission, third party advertisers can place cookies to enable them to show adverts which we think will be relevant to your interests whilst you are on third party websites.

For more information about the cookies we use and your choices regarding cookies, please visit our Cookies Policy or the Cookies section of our Privacy Policy.

Use of Your Personal Data
The Company may use personal data for the following purposes:

• To provide and maintain our service, including to monitor the usage of our Service.

• To manage your Account: to manage your registration as a user of the service. The personal data You provide can give You access to different functionalities of the services that are available to you as a registered user.

• For the performance of a contract: the development, compliance and undertaking of the purchase contract for the products, items or services you have purchased or of any other contract with us through the our services.

• To contact you: To contact you by email, telephone calls, SMS, or other equivalent forms of electronic communication, such as a mobile application’s push notifications regarding updates or informative communications related to the functionalities, products or contracted services, including the security updates, when necessary or reasonable for their implementation.

• To provide you with news, special offers and general information about other goods, services and events which we offer that are similar to those that you have already purchased or enquired about unless You have opted not to receive such information.

• To manage your requests: To attend and manage your requests to us.

• To deliver targeted advertising to you: we may use Your information to develop and display content and advertising (and work with third-party vendors who do so) tailored to your interests and/or location and to measure its effectiveness.

• For business transfers: We may use your information to evaluate or conduct a merger, divestiture, restructuring, reorganization, dissolution, or other sale or transfer of some or all of our assets, whether as a going concern or as part of bankruptcy, liquidation, or similar proceeding, in which personal data held by us about our service’s users are among the assets transferred.

• For other purposes: We may use your information for other purposes, such as data analysis, identifying usage trends, determining the effectiveness of our promotional campaigns and to evaluate and improve our services, products, , marketing and your overall experience.

We may share Your personal information in the following situations:

• With Service Providers: We may share your personal information with service providers to monitor and analyse the use of our services, to show advertisements to you to help support and maintain our services, to advertise on third party websites to you after you visited our website, for payment processing or to contact you.

• For business transfers: We may share or transfer your personal information in connection with, or during negotiations of, any merger, sale of company assets, financing, or acquisition of all or a portion of our business to another company.

• With Affiliates: We may share your information with our affiliates, in which case we will require those affiliates to honour this privacy policy. Affiliates include our parent company and any other subsidiaries, joint venture partners or other companies that we control or that are under common control with us.

• With business partners: We may share your information with our business partners to offer you certain products, services or promotions.

• With other users: when You share personal information or otherwise interact in the public areas with other users, such information may be viewed by all users and may be publicly distributed outside.

• With Your consent: We may disclose your personal information for any other purpose with your consent.

Retention of Your Personal Data
The Company will retain your personal Data only for as long as is necessary for the purposes set out in this privacy policy. We will retain and use your personal data to the extent necessary to comply with our legal obligations (for example, if we are required to retain your data to comply with applicable laws), resolve disputes, and enforce our legal agreements and policies.

The Company will also retain usage data for internal analysis purposes. Usage data is generally retained for a shorter period of time, except when this data is used to strengthen the security or to improve the functionality of our services, or we are legally obligated to retain this data for longer time periods.

How we store your personal data
The General Data Protection Regulation (2018) (GDPR) contains principles affecting employers and other personal records of prospective and current learners/parents, tutors and employees of Shalom Education. Information protected by GDPR includes not only personal data held on a computer, but also certain manual records containing personal data, such as learners/parents and employee personnel files that form part of a structured filing system, database or both.

We ensure that the web server hosting our website and other social media platforms are solely used for the purpose of Shalom Education’s services and not for personal interest. Also, that all personal data that we collected are stored on secured cloud based servers, in accordance with Shalom Education’s data protection obligation, the ICO guideline and for security purposes.

Transfer of Your Personal Data

Your information, including personal data, is processed at the company’s operating offices and in any other places where the parties involved in the processing are located. It means that this information may be transferred to — and maintained on — computers located outside of your state, province, country or other governmental jurisdiction where the data protection laws may differ than those from your jurisdiction.

Your consent to this privacy policy followed by your submission of such information represents your agreement to that transfer.

We will take all steps reasonably necessary to ensure that your data is treated securely and in accordance with this Privacy Policy. Also, we will ensure that no transfer of your personal data will take place to an organisation or a country unless there are adequate controls in place, including the security of your data and other personal information.

Disclosure of Your Personal Data

Business Transactions

If Shalom education Ltd is involved in a merger, acquisition or asset sale, your personal data may be transferred. However, we will provide notice before your personal data is transferred and becomes subject to a different privacy policy.

Law enforcement

Under certain circumstances, the we may be required to disclose your personal data if required to do so by law or in response to valid requests by public authorities (e.g. a court or a government agency).

Other legal requirements

Shalom Education Ltd may disclose your personal data in the good faith belief that such action is necessary to:

• Comply with a legal obligation
• Protect and defend the rights or property of the company
• Prevent or investigate possible wrongdoing in connection with our services
• Protect the personal safety of users of our services or the public
• Protect against legal liability

Security of Your Personal Data

The security of Your Personal Data is important to us, but remember that no method of transmission over the Internet, or method of electronic storage is 100% secure. While we strive to use commercially acceptable means to protect your personal Data, We cannot guarantee its absolute security.

Detailed Information on the Processing of Your Personal Data

The service providers we use may have access to your personal data. These third-party vendors collect, store, use, process and transfer information about your activity on our service in accordance with their privacy policies.

Analytics

We may use third-party Service providers to monitor and analyse the use of our Service. For example:

• Google Analytics
Google Analytics is a web analytics service offered by Google that tracks and reports website traffic. Google uses the data collected to track and monitor the use of our Service. This data is shared with other Google services. Google may use the collected data to contextualise and personalize the ads of its own advertising network.

You can opt-out of having your activity on our website being available to Google Analytics by installing the Google Analytics opt-out browser add-on. The add-on prevents the Google Analytics JavaScript (ga.js, analytics.js and dc.js) from sharing information with Google Analytics about visits activity.

For more information on the privacy practices of Google, please visit the Google Privacy & Terms web page: https://policies.google.com/privacy

Advertising

We may use Service Providers to show advertisements to You to help support and maintain Our Service. For example:

• Google AdSense & DoubleClick Cookie
Google, as a third party vendor, uses cookies to serve ads on our Service. Google’s use of the DoubleClick cookie enables it and its partners to serve ads to our users based on their visit to our website or other websites on the Internet.

You may opt out of the use of the DoubleClick Cookie for interest-based advertising by visiting the Google Ads Settings web page: https://www.google.com/ads/preferences/

Email Marketing

We may use your personal data to contact you with newsletters, marketing or promotional materials and other information that may be of interest to you. You may opt-out of receiving any, or all of these communications from us by following the unsubscribe link or instructions provided in any email we send or by contacting us.

We may use Email Marketing Service Providers to manage and send emails to You. For example, using:

• Mailchimp
Mailchimp is an email marketing sending service provided by The Rocket Science Group LLC.

For more information on the privacy practices of Mailchimp, please visit their Privacy policy: https://mailchimp.com/legal/privacy/

Payments

We provide paid services on our website and we may provide paid products. So, we may use third-party services for payment processing (e.g. payment processors).

We will not store or collect your payment card details. That information is provided directly to our third-party payment processors whose use of your personal information is governed by their privacy policy. These payment processors adhere to the standards set by PCI-DSS as managed by the PCI Security Standards Council, which is a joint effort of brands like Visa, Mastercard, American Express and Discover. PCI-DSS requirements help ensure the secure handling of payment information. We are currently using:

• Stripe
Their Privacy Policy can be viewed at https://stripe.com/us/privacy

Behavioural Remarketing

We use remarketing services to advertise to you after you have accessed or visited our website. We and our third-party vendors use cookies and non-cookie technologies to help us recognize your device and understand how you use our website. Therefore, we can improve our services to reflect your interests and serve you advertisements that are likely to be of more interest to you.

These third-party vendors collect, store, use, process and transfer information about your activity on our website in accordance with their Privacy Policies, which enables us to:

Measure and analyse traffic and browsing activity on our website
Show advertisements for our products and/or services to you on third-party websites or apps
Measure and analyse the performance of our advertising campaigns
Some of these third-party vendors may use non-cookie technologies that may not be impacted by browser settings that block cookies. Your browser may not permit you to block such technologies. Instead, you can use the following third-party tools to decline the collection and use of information for the purpose of serving you interest-based advertising:

The NAI’s opt-out platform: https://www.networkadvertising.org/choices/
The EDAA’s opt-out platform https://www.youronlinechoices.com/
The DAA’s opt-out platform: https://optout.aboutads.info/?c=2&lang=EN
You may opt-out of all personalised advertising by enabling privacy features on your mobile device, such as Limit Ad Tracking (iOS) and Opt Out of Ads Personalisation (Android). Check your mobile device Help system for more information.

We may share information, such as hashed email addresses (if available) or other online identifiers collected on our website with these third-party vendors. This allows our third-party vendors to recognise and deliver you ads across devices and browsers. To read more about the technologies used by these third-party vendors and their cross-device capabilities please refer to the privacy policy of each vendor listed below.

The third-party vendors We use are:

• Google Ads (AdWords)
Google Ads (AdWords) remarketing service is provided by Google Inc.

You can opt-out of Google Analytics for display advertising and customise the Google Display Network ads by visiting the Google Ads Settings page: https://www.google.com/settings/ads

Google also recommends installing the Google Analytics Opt-out Browser Add-on – https://tools.google.com/dlpage/gaoptout – for your web browser. The Google Analytics Opt-out Browser Add-on provides visitors with the ability to prevent their data from being collected and used by Google Analytics.

For more information on the privacy practices of Google, please visit the Google Privacy & Terms web page: https://policies.google.com/privacy

• Facebook
Facebook remarketing service is provided by Facebook Inc.

You can learn more about interest-based advertising from Facebook by visiting this page: https://www.facebook.com/help/516147308587266

To opt-out from Facebook’s interest-based ads, follow these instructions from Facebook: https://www.facebook.com/help/568137493302217

Facebook adheres to the Self-Regulatory Principles for Online Behavioural Advertising established by the Digital Advertising Alliance. You can also opt-out from Facebook and other participating companies through the Digital Advertising Alliance in the USA https://www.aboutads.info/choices/, the Digital Advertising Alliance of Canada in Canada https://youradchoices.ca/ or the European Interactive Digital Advertising Alliance in Europe https://www.youronlinechoices.eu/, or opt-out using your mobile device settings.

For more information on the privacy practices of Facebook, please visit Facebook’s Data Policy: https://www.facebook.com/privacy/explanation

Usage, Performance and Miscellaneous

We may use third-party Service Providers to provide a better improvement of our website. For example, we use:

• Invisible reCAPTCHA
We use an invisible captcha service named reCAPTCHA. which is operated by Google.

The reCAPTCHA service may collect information from you and from your device for security purposes.

The information gathered by reCAPTCHA is held in accordance with the Privacy Policy of Google: https://www.google.com/intl/en/policies/privacy/

GDPR Privacy

Legal Basis for Processing Personal Data under GDPR

We may process personal data under the following conditions:

• Consent: You have given Your consent for processing personal data for one or more specific purposes.

• Performance of a contract: The provision of personal data is necessary for the performance of an agreement with you or for any pre-contractual obligations thereof.

• Legal obligations: Processing personal data is necessary for compliance with a legal obligation to which the company is subject.

• Vital interests: Processing personal data is necessary in order to protect your vital interests or of another natural person.

• Public interests: Processing personal data is related to a task that is carried out in the public interest or in the exercise of official authority vested in the Company.

• Legitimate interests: Processing personal data is necessary for the purposes of the legitimate interests pursued by the Company.

In any case, we will gladly help to clarify the specific legal basis that applies to the processing, and in particular whether the provision of personal data is a statutory (required by law) or a contractual requirement, or a requirement necessary to enter into a contract.

Your Rights under the GDPR

We respect the confidentiality of your personal data and we guarantee that you can exercise your rights.

You have the right under this privacy policy, and by law if you are within the EU or the UK, to:

• Request access to your personal data. The right to access, update or delete the information we have on you. Whenever made possible, you can access, update or request deletion of your personal data directly within your account settings section. If you are unable to perform these actions yourself, please contact us to assist you. This also enables you to receive a copy of the personal data we hold about you.

• Request correction of the personal data that we hold about you. You have the right to have any incomplete or inaccurate information we hold about you corrected.

• Object to processing of your personal data. This right exists where we are relying on a legitimate interest as the legal basis for our processing and there is something about your particular situation, which makes you want to object to our processing of your personal data on this ground. You also have the right to object to us processing your personal data for direct marketing purposes.

• Request erasure of your personal data. You have the right to ask us to delete or remove personal data when there is no good reason for us to continue processing it.

• Request the transfer of your personal data. We will provide to you, or to a third-party you have chosen, your personal data in a structured, commonly used, machine-readable format. Please note that this right only applies to automated information which you initially provided consent for us to use or where we used the information to perform a contract with you.

• Withdraw your consent. You have the right to withdraw your consent on using your personal data. If you withdraw your consent, we may not be able to provide you with access to specific functionalities of our services.

Exercising of Your GDPR Data Protection Rights

You may exercise Your rights of access, rectification, cancellation and opposition by contacting us. Please note that we may ask you to verify your identity before responding to such requests. If you make a request, we will try our best to respond to you as soon as possible.

You have the right to complain to a Data Protection Authority about our collection and use of your personal data. For more information, if you are in the European Economic Area (EEA), please contact Your local data protection authority in the EEA.

Children’s Privacy

Our website may contain content appropriate for children under the age of 13. As a parent, you should know that by using our services, children under the age of 13 may participate in activities that involve the collection or use of personal information. We use reasonable efforts to ensure that before we collect any personal information from a child, the child’s parent receives notice of and consents to our personal information practices.

We may also limit how we collect, use, and store some of the information of users between 13 and 18 years old. In some cases, this means that we will be unable to provide certain functionalities of our services to these users. If we need to rely on consent as a legal basis for processing your information and your country requires consent from a parent, We may require the parent’s consent before we collect and use that information.

We may ask a user to verify their date of birth before collecting any personal information from them. If the user is under the age of 13, the service will either be blocked or redirected to a parental consent process.

Information Collected from Children Under the Age of 13

We may collect and store persistent identifiers, such as cookies or IP addresses from children, without parental consent for the purpose of supporting the internal operation of the our services.

We may collect and store other personal information about children if this information is submitted by a child with prior parental consent or by the parent or guardian of the child.

We may collect and store the following types of personal information about a child when submitted by a child with prior parental consent or by the parent or guardian of the child:

First and/or last name
Date of birth
Gender
Grade level/year group
Email address
Telephone number
Parent’s or guardian’s name
Parent’s or guardian’s email address
For further details on the information we might collect, you can refer to the “Types of Data Collected” section of this privacy policy. We follow our standard privacy policy for the disclosure of personal information collected from and about children.

Parental Access

A parent who has already given the Company permission to collect and use their child’s personal information can, at any time:

Review, correct or delete the child’s personal information
Discontinue further collection or use of the child’s personal information
To make such a request, you can write to us using the contact information provided in this privacy policy.

Links to Other Websites

Our Service may contain links to other websites that are not operated by us. If you click on a third party link, you will be directed to that third party’s site. We strongly advise you to review the privacy policy of every site you visit.

We have no control over and assume no responsibility for the content, privacy policies or practices of any third party sites or services.

Policies Unique to Our Tutoring Services

The following applies to students that have registered for our tutoring services and tutors and employees representing Shalom Education Ltd:

Your profile includes information such as your username and password, purchases or orders made by you, your interests, preferences, and feedback and survey responses.

Booking history including booking requests, subjects booked, set-up test while using our digital resources, any online meeting and recordings of your bookings

Electronic recordings from CCTV, tutoring session, video conference calls, communication history with other users, your interaction with Shalom Education Tutors, records of referral codes shared, redeemed and any promotions that have been applied to your account. All photo’s, videos and conference recordings may be used to protect learner’s, Tutors and employees and may be used as evidence to support any complaint.

Changes to this Privacy Policy

We may update Our Privacy Policy from time to time. We will notify You of any changes by posting the new Privacy Policy on this page.

We may let You know via email and/or a prominent notice on our website prior to the change becoming effective and update the “Last updated” date at the top of this Privacy Policy.

You are advised to review this Privacy Policy periodically for any changes. Changes to this Privacy Policy are effective when they are posted on this page.

Contact Us

If you have any questions about this Privacy Policy, You can contact us:

By email: info@shalom-education.com
By visiting this page on our website: https://www.shalom-education.com/contact
By phone number: 01206657616

I agree

Cookie	Duration	Description
cookielawinfo-checkbox-analytics	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checkbox-functional	11 months	The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checkbox-necessary	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookies is used to store the user consent for the cookies in the category "Necessary".
cookielawinfo-checkbox-others	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checkbox-performance	11 months	This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Performance".
viewed_cookie_policy	11 months	The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. It does not store any personal data.

GCSE Maths

Quadratic Equations

Example

Example

Example

Example

Example

Example

Example

Example – Full exam question

Example – Exam question

Example